Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]. / McKee, James; Smyth, Chris.

In: Algebraic Combinatorics, Vol. 3, No. 3, 02.06.2020, p. 775-789.

Research output: Contribution to journalArticle

E-pub ahead of print


The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices.
In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in Z that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral characterisation of the affine and finite Dynkin diagrams that are not simply laced (much as the graph result gives a spectral characterisation of the simply laced ones).
Original languageEnglish
Pages (from-to)775-789
Number of pages15
JournalAlgebraic Combinatorics
Issue number3
Early online date2 Jun 2020
Publication statusE-pub ahead of print - 2 Jun 2020
This open access research output is licenced under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

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